CS267 Assignment 1
due: January 26, 2000

Computing Pi to 206 billion digits

by François Labelle

Introduction

Although 10 digits are sufficient for all practical purposes, the computation of Pi to as many digits as possible provides an endless challenge. This page describes the September 1999 computation of Pi to 206,158,430,000 decimal digits by Y. Kanada and D. Takahashi of the University of Tokyo. Storing all these digits in ASCII format would require 303 CD-ROMs!

To be able to claim a world record, one must compute Pi using two different methods and (hopefully) get the same answer on both runs. One run can be viewed as the computation and the other as the verification, although the situation is completely symmetrical. Both computations were performed in base-10 throughout since the conversion from base-2 to base-10 would be very expensive.

Details of the Computation

The computer used was the HITACHI SR8000 from the Information Technology Center, Computer Centre Division, University of Tokyo. This computer is currently ranked as the #5 top supercomputer in the world according to this list.

HITACHI SR8000	theoretical speed	memory used
1 node	8 Gflop/s	6.758 GB
all 128 nodes	1 Tflop/s	865 GB

run	algorithm	time taken (h:m:s)
1.	Gauss-Legendre algorithm	37:21:04
2.	Borwein's 4-th order convergent algorithm	46:07:10

Successfully parallelized?

The researchers claim that all 128 processing elements were "definitely" used. It's hard to verify the claim, as very few details are given by the researchers.

According to Stu's pi page, the fastest PC program to compute Pi is PiFast. By interpolating the time taken by PiFast to compute 1 million and 64 million digits in core on a Pentium 450MHz, I get the following formula:

time = 2.40e-10 * digit^1.253 hours

By extrapolating, I estimate that computing 206 billion digits should take 36000 hours (4.1 years) on the same Pentium assuming an unbounded main memory. The supercomputer run was about 1000 times faster than that. This is enough to convince me that the computation was efficiently carried.

Moore's Law for the digits of Pi

Following the computer performance trend, the number of known digits of Pi appears to grow exponentially with time, as shown on the following semi-log graph. Are we going to reach 10¹⁴ digits of Pi by the year 2010? :)

based on this Brief History of Pi Calculation with Computers